Finding max/min of 3d graphs with chain rule

[tufor]

http://math.berkeley.edu/~theojf/Midterm2Practice.pdf

can someone please help me on problem number 2 of the link above?
apologies for the bad handwriting. my professor is just horrible with that.

i've done max and min with multivariables before and I've done chain rule , but I've never combined the two. also, in that problem, the only critical point seems to be a saddle point, so I'm not sure what the "lowest" point is referring to

 

[HallsofIvy]

You don't need to use the chain rule- in fact you don't need to worry about multi-variables at all. Replace x and y in z(x,y) with their expressions in terms of t and differentiate with respect to t. There are two critical points. One is a minimum and the other a maximum.